On the relationships between types of \(L\)-convergence spaces (Q2805881)

If \(L\) is a complete residuated lattice, then a stratified \(L\)-filter on \(X\) is a mapping \(\mathcal F : L ^X \to L\) that preserves meets, maps 0 to 0 and 1 to 1 and has the property \(\mathcal F(\alpha * \lambda) \geq \alpha * \mathcal F(\lambda)\) for all \(\alpha \in L, \lambda \in L^X\). The set of all stratified \(L\) filters is denoted by \(\mathcal F_L^s(X)\). If to each \(\alpha \in L\) there is a mapping \(q_{\alpha}: \mathcal F_L^s(X) \to 2^X\) fulfilling some natural properties, then the collection of all such mappings is called a stratified levelwise \(L\)-convergence structure on \(X\). The set \(X\) with such collection is a stratified levelwise \(L\)-convergence space. A function from \( \mathcal F_L^s(X)\) to \(L^X\) with two additional properties is a stratified \(L\)-convergence structure or (if it preserves the order) a strong stratified \(L\)-convergence structure. The mutual relations among the corresponding spaces are studied. It is shown that a stratified strong \(L\)-convergence space coincides with a strong left-continuous stratified \(L\)-convergence space and a condition for a stratified \(L\)-convergence space to be a strong one is found.